where we have used the fact that \(\mathcal {S}^2_{\mu _1,\mu _2,\beta }< \min \lbrace \frac{1}{\mu _1},\frac{1}{\mu _2}\rbrace  \mathcal {S}^2\) . By direct calculations, there exists \(0\!<\!\hat{\rho }<R_0\)  such that \(h_1(\hat{\rho })=0\) .
In [1]}, [2]}, it has been proved that
\(m^+(a_1,0)=\inf _{u\in \mathcal {T}^+_{a_1,p,\mu _1,\alpha _1}} \mathcal {A}_{p,\mu _1,\alpha _1}(u)=\inf _{u\in S(a_1)\cap B_{\hat{\rho }}}\mathcal {A}_{p,\mu _1,\alpha _1}(u).\) 
